X₅₇₃

ORTHOHARMONIC OF X₅₈

X₅₇₃ = inverse-in-Brocard-circle of X₅₇₂

5, 6, 7 3, 4, 5 Kepler Kimberling

X₅₇₃ (ORTHOHARMONIC OF X₅₈) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {991, 3}

• Related Centers: X₉₉₁ (INTERSECTION OF LINES X), X₃ (CIRCUMCENTER)

Trilinears: \(\cos(A + U) : :\), where cot \(U = \cot(A/2) \cot(B/2) \cot(C/2)\) (Joe Goggins, 11/26/08)

Search Value (6, 9, 13): \(10.107230280042\)

▸ Alternate Coordinates3 forms

• Trilinears: \((c - a)\cos B \sin(A - B) - (a - b)\cos C \sin(C - A) : :\)

• Trilinears: \(a[a^{2}(b + c) - abc - b^{3} - c^{3}] : :\)

• Trilinears: s cos A - r sin A, s = semiperimeter, r = inradius

▸ Linear Combinations1 combination

• X₅₇₃ = \(s\) X₃ - \(r \cot(\omega )\) X₆

▸ Incident Central Lines16 lines

{1, 941}, {3, 6}, {4, 9}, {10, 5179}, {20, 391}, {36, 604}, {37, 517}, {43, 165}, {51, 1011}, {55, 181}, {101, 102}, {109, 478}, {184, 199}, {256, 981}, {346, 1018}, {347, 1020}

▸ Functional Relations12 relations

• X₅₇₃ = reflection of X₉₉₁ in X₃

• X₅₇₃ = isogonal conjugate of X₁₃₄₇₈

• X₅₇₃ = inverse-in-Brocard-circle of X₅₇₂

• X₅₇₃ = X₃₃₃-Ceva conjugate of X₁

• X₅₇₃ = crosspoint of X₅₉ and X₁₉₀

• X₅₇₃ = crosssum of X₁₁ and X₆₄₉

• X₅₇₃ = crossdifference of every pair of points on line X₅₂₃X₁₄₅₉

• X₅₇₃ = X₂₁₆-of-excentral-triangle

• X₅₇₃ = perspector, wrt excentral triangle, of the excentral-hexyl ellipse

• X₅₇₃ = inverse-in-excircles-radical-circle of X₅₁₇₉

• X₅₇₃ = Cundy-Parry Phi transform of X₅₈

• X₅₇₃ = Cundy-Parry Psi transform of X₁₀

Notes

X₅₇₃ is the internal center of similitude of the circumcircle and Apollonius circle. The external center is X₃₈₆. (Peter J. C. Moses, 8/22/03)

Let A' be the inverse-in-excircles-radical-circle of the midpoint of BC, and define B' and C' cyclically. Triangle A'B'C' is inscribed in the Apollonius circle and homothetic to the 1st circumperp triangle at X₅₇₃. (Randy Hutson, December 2, 2017)